The union of independent USFs on Zd is transient

Eleanor Archer; Asaf Nachmias; Matan Shalev; Pengfei Tang

doi:10.1214/24-ECP609

2024 The union of independent USFs on

Z^{d}

is transient

Eleanor Archer, Asaf Nachmias, Matan Shalev, Pengfei Tang

Author Affiliations +

Electron. Commun. Probab. 29: 1-8 (2024). DOI: 10.1214/24-ECP609

Abstract

We show that the union of two or more independent uniform spanning forests (USF) on $Z^{d}$ with $d \geq 3$ almost surely forms a connected transient graph. In fact, this also holds when taking the union of a deterministic everywhere percolating set and an independent ε-Bernoulli percolation on a single USF sample.

Funding Statement

This research is supported by the ERC consolidator grant 101001124 (UniversalMap) as well as ISF grants 1294/19 and 898/23.

Acknowledgments

We would like to thank Peleg Michaeli for posing the question of Theorem 1.2. We also thank him, Matan Harel and Ofir Karin for useful discussions, and the anonymous referee for a careful reading and comments which greatly simplified the proof.

Citation

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Eleanor Archer. Asaf Nachmias. Matan Shalev. Pengfei Tang. "The union of independent USFs on $Z^{d}$ is transient." Electron. Commun. Probab. 29 1 - 8, 2024. https://doi.org/10.1214/24-ECP609